Question

a student places a metal ball at point a and lets it roll down the track to point z as shown below. which of the following is not true about the amount of potential energy of the ball as it moves down the track? the ball has the most potential energy at a the ball will have more potential energy at x than at z the ball will have more potential energy at y than at z the ball will have more potential energy at w than at z

Explanation:

Step1: Recall the formula for gravitational potential energy

The formula for gravitational potential energy is \(U = mgh\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (\(g= 9.8\space m/s^{2}\)), and \(h\) is the height of the object above a reference level. Since the mass \(m\) and \(g\) are constant for the ball, the potential energy depends only on the height \(h\).

Step2: Compare the heights at different points

At point \(A\), the height \(h_A=1.5\space m\). At point \(W\), \(h_W = 1\space m\). At point \(X\), \(h_X=1.2\space m\). At point \(Y\), \(h_Y = 0.7\space m\). At point \(Z\), \(h_Z=0.9\space m\).

• For the statement “The ball has the most potential energy at \(A\)”: Since \(h_A = 1.5\space m\) is the maximum height among all points (\(1.5>1.2 > 1>0.9>0.7\)), \(U_A=mgh_A\) is the maximum. This statement is true.
• For the statement “The ball will have more potential energy at \(X\) than at \(Z\)”: Since \(h_X = 1.2\space m\) and \(h_Z=0.9\space m\), and \(U = mgh\), \(U_X=mgh_X\) and \(U_Z=mgh_Z\). Since \(h_X>h_Z\), \(U_X > U_Z\). This statement is true.
• For the statement “The ball will have more potential energy at \(Y\) than at \(Z\)”: Since \(h_Y=0.7\space m\) and \(h_Z = 0.9\space m\), and \(U=mgh\), \(U_Y=mgh_Y\) and \(U_Z=mgh_Z\). Since \(h_Y < h_Z\), \(U_Y
• For the statement “The ball will have more potential energy at \(W\) than at \(Z\)”: Since \(h_W = 1\space m\) and \(h_Z=0.9\space m\), and \(U = mgh\), \(U_W=mgh_W\) and \(U_Z=mgh_Z\). Since \(h_W>h_Z\), \(U_W>U_Z\). This statement is true.

Answer:

The ball will have more potential energy at \(Y\) than at \(Z\)